Synthesis of Polynomic Systems

With H a Hilbert space and $\{ (x_i ,y_i ):i = 1, \cdots ,m\} \subset H \times H$ a basic problem is to determine the existence and uniqueness of causal functions, f, on H satisfying $y_i = fx_i $$i = 1, \cdots ,m$. The present paper considers classes of polynomic functions which minimize an operator norm. The results include explicit necessary and sufficient conditions and an explicit synthesis procedure for realizing the resultant polynomic functions.